A complex number $z_1$ has a magnitude $|z_1|=12$ and an angle $\theta_1=45^{\circ}$. Express $z_1$ in rectangular form, as $z_1=a+bi$. Express $a+bi$ in exact terms. $z_1 = $
Explanation: The Strategy A complex number of the form $z={a}+{b}i$ has: A magnitude of ${|z|}=\sqrt{{a}^2+{b}^2}$. An angle of ${\theta}=\arctan\left(\dfrac{{b}}{{a}}\right)$. [How did we get these equations?] Therefore, given the absolute value ${|z|}$ and angle ${\theta}$, the parts ${a}$ and ${b}$ are given by the following two equations: ${a}={|z|}\cos{\theta}$ ${b}={|z|}\sin{\theta}$ [How did we get these equations?] Finding $a$ For ${|z_1|}={12}$ and ${\theta_1}={45^{\circ}}$, we can find ${a}$ as follows. $\begin{aligned}{a}&={|z_1|}\cos{\theta_1} \\\\&={12}\cos{45^\circ} \\\\&={6\sqrt{2}}\end{aligned}$ Finding $b$ $\begin{aligned}{b}&={|z_1|}\sin{\theta_1} \\\\&={12}\sin{45^\circ} \\\\&={6\sqrt{2}}\end{aligned}$ Summary We found that ${a}={6\sqrt{2}}$ and ${b}={6\sqrt{2}}$. Therefore, $z_1=6\sqrt{2}+6\sqrt{2}i$.